Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

For the hypergeometric spectral curve and its confluent degenerations (spectral curves of "hypergeometric type"), we obtain a simple formula expressing the topological recursion free energies as a sum over BPS states (degenerate spectral networks) for a corresponding quadratic differential $\varphi$. In doing so, we generalize Gaiotto-Moore-Neitzke's construction of BPS structures to include the case where $\varphi$ has simple poles or supports a degenerate ring domain. For the nine spectral curves of hypergeometric type, we provide a complete description of the corresponding BPS structures over a generic locus in the relevant parameter space; in particular, we prove the existence of saddles trajectories at the expected parameter values. We determine the corresponding BPS cycles, central charges, and BPS invariants, and verify our formula in each case. We conjecture that a similar relation should hold more generally whenever the corresponding BPS structure is uncoupled, and provide experimental evidence in two simple higher rank examples.